This series has the same property as the Fibonacci series in that each term is the sum of the two before it. It is called the Lucas series, after Édouard Lucas (1842-1891), a French mathematician who extensively studied both series. He also invented the Tower of Hanoi puzzle.

n

T(n)

2n

T(2n)

T(2n)

T(n)

3n

T(3n)

T(3n)

T(n)

4n

T(4n)

T(4n)

T(n)

1

1

2

1

1

3

2

2

4

3

3

2

1

4

3

3

6

8

8

8

21

21

3

2

6

8

4

9

34

17

12

144

72

4

3

8

21

7

12

144

48

16

987

329

5

5

10

55

11

15

610

122

20

6765

1353

6

8

12

144

18

18

2584

323

24

46368

5796

7

13

14

377

29

21

10946

842

28

317811

24447

8

21

16

987

47

24

46368

2208

32

2178309

103729

9

34

18

2584

76

27

196418

5777

36

14930352

439128

10

—

55

——

20

————

6765

———

123

——

30

————

832040

———

15128

———

40

————

102334155

—————

1860621

————

The figures in the T(2n) / T(n) column show a number series with the same property as the Fibonacci: each number in the series is the sum of the two before it. However, instead of beginning 0, 1,… it begins 2, 1,…, with 2 instead of 0 as the 0th term. This series is called the Lucas series. Terms 0 to 40 of both series are shown in the table below.